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Theorem Quotes (46 quotes)

...durch planmässiges Tattonieren.
(... through systematic, palpable experimentation.)
Response, when asked how he came upon his theorems.
Quoted in A.L. Mackay, Dictionary of Scientific Quotations (1994).
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A distinguished Princeton physicist on the occasion of my asking how he thought Einstein would have reacted to Bell’s theorem. He said that Einstein would have gone home and thought about it hard for several weeks … He was sure that Einstein would have been very bothered by Bell’s theorem. Then he added: “Anybody who’s not bothered by Bell’s theorem has to have rocks in his head.”
In 'Is the Moon There When Nobody Looks? Reality and the Quantum Theory', Physics Today (Apr 1985), 38-47.
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A mathematician is a device for turning coffee into theorems.
In D. S. Richeson, Euler's Gem: The Polyhedron Formula and the Birth of Topology (2008), Preface, ix. Also attributed incorrectly to Paul Erdos, who did often repeat it.
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An applied mathematician loves the theorem. A pure mathematician loves the proof.
Anonymous
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And I believe that the Binomial Theorem and a Bach Fugue are, in the long run, more important than all the battles of history.
This Week Magazine (1937).
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Bell’s theorem is easy to understand but hard to believe.
…...
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But neither thirty years, nor thirty centuries, affect the clearness, or the charm, of Geometrical truths. Such a theorem as “the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the sides” is as dazzlingly beautiful now as it was in the day when Pythagoras first discovered it, and celebrated its advent, it is said, by sacrificing a hecatomb of oxen—a method of doing honour to Science that has always seemed to me slightly exaggerated and uncalled-for. One can imagine oneself, even in these degenerate days, marking the epoch of some brilliant scientific discovery by inviting a convivial friend or two, to join one in a beefsteak and a bottle of wine. But a hecatomb of oxen! It would produce a quite inconvenient supply of beef.
Written without pseudonym as Charles L. Dodgson, in Introduction to A New Theory of Parallels (1888, 1890), xvi. Note: a hecatomb is a great public sacrifice, originally of a hundred oxen.
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Chebyshev said, and I say it again. There is always a prime between n and 2n.
[Referring to the theorem known as the Bertrand postulate, partially solved by Joseph Bertrand (1822-1900). Chebyshev produced a complete, but difficult, proof. Paul Erdös gave an elegant new proof at age 18, while a second-year undergraduate.] Rhyme quoted by Béla Bollobás, 'The Life and Work of Paul Erdos", in Shiing-Shen Chern and Friedrich Hirzebruch (eds.) Wolf Prize in Mathematics (2000), Vol. 1, 296.
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Everyone believes in the normal law, the experimenters because they imagine that it is a mathematical theorem, and the mathematicians because they think it is an experimental fact.
Conversation with Henri Poincaré. In Henri Poincaré, Calcul des Probabilités (1896), 171.
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Gradually, at various points in our childhoods, we discover different forms of conviction. There’s the rock-hard certainty of personal experience (“I put my finger in the fire and it hurt,”), which is probably the earliest kind we learn. Then there’s the logically convincing, which we probably come to first through maths, in the context of Pythagoras’s theorem or something similar, and which, if we first encounter it at exactly the right moment, bursts on our minds like sunrise with the whole universe playing a great chord of C Major.
In short essay, 'Dawkins, Fairy Tales, and Evidence', 2.
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I approached the bulk of my schoolwork as a chore rather than an intellectual adventure. The tedium was relieved by a few courses that seem to be qualitatively different. Geometry was the first exciting course I remember. Instead of memorizing facts, we were asked to think in clear, logical steps. Beginning from a few intuitive postulates, far reaching consequences could be derived, and I took immediately to the sport of proving theorems.
Autobiography in Gösta Ekspong (ed.), Nobel Lectures: Physics 1996-2000 (2002), 115.
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I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply the notes of our observations.
In A Mathematician's Apology (1940, reprint with Foreward by C.P. Snow 1992), 113.
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I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.
Recalling the degree of focus and determination that eventually yielded the proof of Fermat's Last Theorem.
Quoted in interview for PBS TV program Nova. In William Byers, How Mathematicians Think (2007), 1.
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I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths.
Grundgesetze der Arithmetik (1893), xiii, trans. Ivor Grattan-Guinness.
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I confess that Fermat's Theorem as an isolated proposition has very little interest for me, for a multitude of such theorems can wasily be set up, which one could neither prove nor disprove. But I have been stimulated by it to bring our again several old ideas for a great extension of the theory of numbers. Of course, this theory belongs to the things where one cannot predict to what extent one will succeed in reaching obscurely hovering distant goals. A happy star must also rule, and my situation and so manifold distracting affairs of course do not permit me to pursue such meditations as in the happy years 1796-1798 when I created the pricipal topics of my Disquisitiones arithmeticae. But I am convinced that if good fortune should do more than I expect, and make me successful in some advances in that theory, even the Fermat theorem will appear in it only as one of the least interesting corollaries.
In reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem. The hope Gauss expressed for his success was never realised.
Letter to Heinrich Olbers (21 Mar 1816). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 413.
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I had a dislike for [mathematics], and ... was hopelessly short in algebra. ... [One extraordinary teacher of mathematics] got the whole year's course into me in exactly six [after-school] lessons of half an hour each. And how? More accurately, why? Simply because he was an algebra fanatic—because he believed that algebra was not only a science of the utmost importance, but also one of the greatest fascination. ... [H]e convinced me in twenty minutes that ignorance of algebra was as calamitous, socially and intellectually, as ignorance of table manners—That acquiring its elements was as necessary as washing behind the ears. So I fell upon the book and gulped it voraciously. ... To this day I comprehend the binomial theorem.
In Prejudices: third series (1922), 261-262.
For a longer excerpt, see H. L. Mencken's Recollections of School Algebra.
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If all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime.
In When You Were a Tadpole and I Was a Fish: And Other Speculations About This and That (), 124.
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If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.
In How to Solve It: A New Aspect of Mathematical Method (2004), 15.
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It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.
Quoted in Mark Kac, Statistical Independence in Probability, Analysis and Number Theory (1959), 52.
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It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, chusing [choosing] rather to acknowledge their ignorance, than affirm anything rashly. They affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour, rejecting all probable conjectures and little witticisms. They submit nothing to authority, indulge no affection, detest subterfuges of words, and declare their sentiments, as in a Court of Judicature [Justice], without passion, without apology; knowing that their reasons, as Seneca testifies of them, are not brought to persuade, but to compel.
Mathematical Lectures (1734), 64.
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It was a dark and stormy night, so R. H. Bing volunteered to drive some stranded mathematicians from the fogged-in Madison airport to Chicago. Freezing rain pelted the windscreen and iced the roadway as Bing drove on—concentrating deeply on the mathematical theorem he was explaining. Soon the windshield was fogged from the energetic explanation. The passengers too had beaded brows, but their sweat arose from fear. As the mathematical description got brighter, the visibility got dimmer. Finally, the conferees felt a trace of hope for their survival when Bing reached forward—apparently to wipe off the moisture from the windshield. Their hope turned to horror when, instead, Bing drew a figure with his finger on the foggy pane and continued his proof—embellishing the illustration with arrows and helpful labels as needed for the demonstration.
In 'R. H. Bing', Biographical Memoirs: National Academy of Sciences (2002), 49. Anecdote based on the recollections of Bing's colleagues, Steve Armentrout and C. E. Burgess. The narrative was given in a memorial tribute at the University of Texas at Austin.
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Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experiment and guesswork.
I Want to be a Mathematician: an Automathography in Three Parts (1985), 321.
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Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems—general and specific statements—can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
Essay,'Private Games', in Lewis Wolpert, Alison Richards (eds.), A Passion for Science (1988), 53.
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No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful.
As quoted in Desmond MacHale. Comic Sections (1993), 107, without citation. Please contact the Webmaster if you know the primary source.
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Nothing in our experience suggests the introduction of [complex numbers]. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius.
In 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences,' Communications in Pure and Applied Mathematics (Feb 1960), 13, No. 1 (February 1960). Collected in Eugene Paul Wigner, A.S. Wightman (ed.), Jagdish Mehra (ed.), The Collected Works of Eugene Paul Wigner (1955), Vol. 6, 537.
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One evening at a Joint Summer Research Congerence in the early 1990’s Nicholai Reshetikhin and I [David Yetter] button-holed Flato, and explained at length Shum’s coherence theorem and the role of categories in “quantum knot invariants”. Flato was persistently dismissive of categories as a “mere language”. I retired for the evening, leaving Reshetikhin and Flato to the discussion. At the next morning’s session, Flato tapped me on the shoulder, and, giving a thumbs-up sign, whispered, “Hey! Viva les categories! These new ones, the braided monoidal ones.”
In David N. Yetter, Functorial Knot Theory: Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants (2001), 8. Yetter writes this personal anecdote is given as a narrative in his own words. Presumable the phrases in quotation marks are based on recollection when written years later.
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The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic.
Remarking on the paper, ‘Statistics in Agricultural Research’ by J. Wishart, Journal of the Royal Statistical Society, Supplement (1934), 1, 52. As cited in Michael Cowle, Statistics in Psychology: An Historical Perspective (2005), 210.
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The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.
In Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (1981). As cited, with no more details, in Yi Ma, An Invitation to 3-D Vision (2004), 228.
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The fact that the proof of a theorem consists in the application of certain simple rules of logic does not dispose of the creative element in mathematics, which lies in the choice of the possibilities to be examined.
As co-author with Herbert Robbins, in What Is Mathematics?: An Elementary Approach to Ideas and Methods (1941, 1996), 15.
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The presentation of mathematics where you start with definitions, for example, is simply wrong. Definitions aren't the places where things start. Mathematics starts with ideas and general concepts, and then definitions are isolated from concepts. Definitions occur somewhere in the middle of a progression or the development of a mathematical concept. The same thing applies to theorems and other icons of mathematical progress. They occur in the middle of a progression of how we explore the unknown.
Interview for website of the Mathematical Association of America.
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The primes are the raw material out of which we have to build arithmetic, and Euclid’s theorem assures us that we have plenty of material for the task.
In A Mathematician's Apology (1940, 2012), 99.
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The results of mathematics are seldom directly applied; it is the definitions that are really useful. Once you learn the concept of a differential equation, you see differential equations all over, no matter what you do. This you cannot see unless you take a course in abstract differential equations. What applies is the cultural background you get from a course in differential equations, not the specific theorems. If you want to learn French, you have to live the life of France, not just memorize thousands of words. If you want to apply mathematics, you have to live the life of differential equations. When you live this life, you can then go back to molecular biology with a new set of eyes that will see things you could not otherwise see.
In 'A Mathematician's Gossip', Indiscrete Thoughts (2008), 213.
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The scientist has to take 95 per cent of his subject on trust. He has to because he can't possibly do all the experiments, therefore he has to take on trust the experiments all his colleagues and predecessors have done. Whereas a mathematician doesn't have to take anything on trust. Any theorem that's proved, he doesn't believe it, really, until he goes through the proof himself, and therefore he knows his whole subject from scratch. He's absolutely 100 per cent certain of it. And that gives him an extraordinary conviction of certainty, and an arrogance that scientists don't have.
Essay, 'Private Games', in Lewis Wolpert and Alison Richards, (eds.), A Passion for Science (1988), 61.
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The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.
The original script by Noel Langley was revised by Florence Ryerson and ‎Edgar Allan Woolf to produce the final screenplay of movie The Wizard of Oz. Whoever originated this particular line, it was said by the Scarecrow character on receiving his Doctor of Thinkology Degree from the Wizard. Screenplay as in The Wizard of Oz: The Screenplay (1989), 123.
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The theory that gravitational attraction is inversely proportional to the square of the distance leads by remorseless logic to the conclusion that the path of a planet should be an ellipse .... It is this logical thinking that is the real meat of the physical sciences. The social scientist keeps the skin and throws away the meat.... His theorems no more follow from his postulates than the hunches of a horse player follow logically from the latest racing news. The result is guesswork clad in long flowing robes of gobbledygook.
In Science is a Sacred Cow (1950), 149-50.
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The traditional method of confronting the student not with the problem but with the finished solution means depriving him of all excitement, to shut off the creative impulse, to reduce the adventure of mankind to a dusty heap of theorems.
In The Act of Creation (1964), 266.
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The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.
In 'A Story With a Moral', Mathematical Gazette (Jun 1973), 57, No. 400, 87.
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The “seriousness” of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects.
In A Mathematician's Apology (1940, 2012), 89.
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Theorem proving is seductive—and its Lorelei voices can put us on the rocks.
In 'Sunset Salvo', The American Statistician (Feb 1986), 40, No. 1, 74.
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Theorems are not to mathematics what successful courses are to a meal.
In Rota's 'Introduction' written (1980) to preface Philip J. Davis and Reuben Hersh, The Mathematical Experience (1981, 2012), xxii-xxiii.
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There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general and it is extremely peculiar that such a procedure has led to so few of the so-called paradoxes.
From letter to his professor Christoffer Hansteen (1826) in Oeuvres, 2, 263-65. In Morris Kline, Mathematical Thought from Ancient to Modern Times (1990), Vol. 3, 947.
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There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.
In George Pólya and Gerald L. Alexanderson (ed.) The Pólya Picture Album: Encounters of a Mathematician (1987), 154. Also footnoted in Matti Tedre, The Development of Computer Science: a Sociocultural Perspective (2006), 198, cited as from How to Solve It: A New Aspect of Mathematical Method (1957), xv.
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We often hear that mathematics consists mainly of “proving theorems.” Is a writer's job mainly that of “writing sentences?”
In Rota's 'Introduction' written (1980) to preface Philip J. Davis and Reuben Hersh, The Mathematical Experience (1981, 2012), xxii.
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What we do may be small, but it has a certain character of permanence and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.
From Inaugural Lecture, Oxford (1920). Recalled in A Mathematician’s Apology (1940, 1967), 76.
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Young men should prove theorems, old men should write books.
Quoted by Freeman Dyson as the answer from G.H. Hardy about the book An Introduction to the Theory of Numbers. Dyson, while at Cambridge, had asked him why “he spent so much time and effort writing that marvellous book when he might be doing serious mathematics.” In Freeman Dyson 'A Walk Through Ramanujan's Garden', Lecture by Dyson at Ramanujan Centenary Conference (2 Jun 1987). Collected in Selected Papers of Freeman Dyson with Commentary (1996), 189. Also as quoted in 'Mathematician, Physicist, and Writer.' Interview (Jun 1990) with Donald J. Albers, The College Mathematics Journal (Jan 1994), 25, No. 1, 4.
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[My favourite fellow of the Royal Society is the Reverend Thomas Bayes, an obscure 18th-century Kent clergyman and a brilliant mathematician who] devised a complex equation known as the Bayes theorem, which can be used to work out probability distributions. It had no practical application in his lifetime, but today, thanks to computers, is routinely used in the modelling of climate change, astrophysics and stock-market analysis.
Quoted in Max Davidson, 'Bill Bryson: Have faith, science can solve our problems', Daily Telegraph (26 Sep 2010)
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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